Temporal Resolution of Uncertainty and Dynamic Choice Theory David M.Kreps Evan L.Porteus Econometrica(Jan.1978) KINOSHITA Takuya

Difference between previously and this paper Payoff vector approach : von NeumanMorgenstern utility function More general

Axiomatic approach : this paper Temporal resolution of uncertainty Uncertainty is “dated”by the time of its resolution

Individual regards uncertainties resolving at different times as being different

For example 1

A fair coin is to be flipped If it comes up heads (𝑧0, 𝑧1 )=(5,10)

If it comes up tail (𝑧0, 𝑧1 )=(5,0)

For someone who has cardinal utility on the vector of payoffs , it will not matter when the flip occurs

Preliminary

T : time , a finite integer t=0,1,…..,T 𝒁𝒕 :a set of possible payoffs(complete separable metric) 𝒀𝒕 :a set of payoff histories up to time t 𝑌1 =𝑍0 , 𝑌𝑡 =𝑌𝑡−1 ×𝑍𝑡−1 𝑫𝒕 :the set of actions(Borel probability measures on 𝑍𝑡 ) 𝐷𝑡−1 =𝑍𝑡−1 ×𝑋𝑡 (endowed with Prohorov metric) 𝑿𝒕 :the set of states(nonempty closed subsets of 𝐷𝑡 )( endowed with Hausdorff metric)

1.Definitions

Definitions :A dynamic choice problem (over{𝑍𝑡 })from time t to T is any element 𝑥𝑡 of 𝑋𝑡 .An action at time t is any element 𝑑𝑡 of 𝐷𝑡 determines

𝑑𝑡

Probability distribution

( 𝑍𝑡 , 𝑥𝑡+1 )

For example 2

(pp188)

2.Choice behavior at a point in time AXIOM 2.1 :For each t and 𝑦𝑡 ,the individual’s choices from closed subsets of 𝐷𝑡 are representable by a complete and transitive binary relation ≽𝑦𝑡 on 𝐷𝑡 . (independence of irrelevant alternatives) AXIOM2.2:For each t and 𝑦𝑡 , ≽𝑦𝑡 ,is continuous (Continuity)

2.Choice behavior at a point in time

AXIOM2.3:For each t and 𝑦𝑡 ,if d , d’ ∈ 𝐷𝑡 are such that d ≽𝑦𝑡 d’ , then(α;d , d’’) ≽𝑦𝑡 (α;d’,d’’) for all α ∈(0,1) and for all d’’ ∈ 𝐷𝑡 (substitution) LEMMA3:Axioms 2.1 2.2 2.3 are necessary and sufficient for there to exist, for each 𝑦𝑡 ,a(bounded)continuous function 𝑈𝑦𝑡 : 𝑍𝑡 × 𝑋𝑡+1 →R such that for d , d’ ∈ 𝐷𝑡 ,d ≽𝑦𝑡 d’ if and only if 𝐸𝑑 [𝑈𝑦𝑡 ] ≽ 𝐸𝑑′ [𝑈𝑦𝑡 ]

3.Temporal Consistency and the Representation Theorem AXIOM3.1:For all t , y ∈ 𝑌𝑡 , z ∈ 𝑍𝑡 ,and x , x’ ∈ 𝑋𝑡+1 , (z , x) ≽𝑦𝑡 (z , x’) at time t if and only if x ≽(𝑦,𝑧) x’ at time t+1 (temporal consistency)

3.Temporal Consistency and the Representation Theorem LEMMA4:Axioms2.1 2.2 2.3 3.1are necessary and sufficient for there to exist functions 𝑈𝑦𝑡 , as in LEMMA3 and ,for fixed{𝑈𝑦𝑡 },unique functions 𝑢𝑦𝑡 :{(z,γ) ∈ 𝑍𝑡 ×R : γ=𝑈(𝑦𝑡 ,𝑧) (x)for some x ∈ 𝑋𝑡+1 }→R Which are strictly increasing in their second argument and which satisfy (1) 𝑈𝑦 (z , x)=𝑢𝑦 (z , 𝑈(𝑦𝑡,𝑧) (x)) for all y ∈ 𝑌𝑡 , z ∈ 𝑍𝑡 and x ∈ 𝑋𝑡+1 (2) 𝑈𝑦 (z , x)=𝑢𝑦 (z , max𝐸𝑑 [𝑈(𝑦,𝑧) ] ) =max 𝑢𝑦 (z , 𝐸𝑑 [𝑈(𝑦,𝑧) ] ) ((2) is alternative form of (1))

3.Temporal Consistency and the Representation Theorem

Theorem1:Axioms2.1 2.2 2.3 3.1 are necessary and sufficient for there to exist a continuous function U : 𝑌𝑇+1 →R and for t=0,…,T－1,continuous functions 𝑢𝑡 : 𝑌𝑡 ×𝑍𝑡 ×R →R, strictly increasing in their third argument , so that if we define 𝑈𝑦𝑇 (𝑧𝑇 )= 𝑈 (𝑦𝑇 ,𝑧𝑇 ) and , recursively, 𝑈𝑦𝑡 (𝑧𝑡 , 𝑥𝑡+1 )=max 𝑢𝑡 (𝑦𝑡 , 𝑧𝑡 , 𝐸𝑑 [𝑈(𝑦𝑡 ,𝑧𝑡 ) ] ) Then for all 𝑦𝑡 and d , d’ ∈ 𝐷𝑡 , d ≽𝑦𝑡 d’ if and only if 𝐸𝑑 [𝑈𝑦𝑡 ] ≽ 𝐸𝑑′ [𝑈𝑦𝑡 ]

4.Temporal Resolution of Uncertainty and Temporal Lotteries

Suppose that

𝑢0 (𝑧0 ,γ) =𝛾 1/2 𝐸𝑑0(𝑎) [𝑈(𝑦0) ]=1.748 𝐸𝑑0(𝑏) [𝑈(𝑦0) ]=1.732

4.Temporal Resolution of Uncertainty and Temporal Lotteries

Definitions : Elements of 𝐷0∗ are called temporal lotteries. Elements of 𝑃𝑡 (𝑦𝑡 )(for any t and 𝑦𝑡 )are called temporal lotteries resolving from time t

4.Temporal Resolution of Uncertainty and Temporal Lotteries

AXIOM4.1 : The relation ≽ is complete and transitive on 𝐷0∗ AXIOM4.2 : The relation ≽ is continuous on 𝐷0∗ AXIOM4.3 : If p , p’ ∈ 𝑃𝑡 (𝑦𝑡 ) satisfy p≻p’, then (t , α ; p , p’’) ≻ (t , α ; p’ , p’’) for all α ∈(0,1) and p’’ ∈ 𝑃𝑡 (𝑦𝑡 ) (Temporal substitution)

4.Temporal Resolution of Uncertainty and Temporal Lotteries Theorem2 : The existence of a relation ≽ on 𝐷𝑡∗ satisfying Axioms 4.1 4.2 4.3 is necessary and sufficient for there to exist continuous functions 𝑈 ∗ : 𝑌𝑇+1 →R and 𝑢𝑡∗ : 𝑌𝑡 ×𝑍𝑡 ×R →R (t=0,…,T－1) such that (i)each 𝑢𝑡∗ is strictly increasing in its third argument , ∗ ∗ (ii) if one defines 𝑈𝑦𝑇 : 𝑍𝑇 →R by 𝑈𝑦𝑇 (𝑍𝑇 )= 𝑈 ∗ (𝑦𝑇 , 𝑧𝑇 )and , recursively , ∗ ∗ 𝑈𝑦𝑡 : 𝑍𝑡 × 𝑋𝑡+1 →R by ∗ (3) 𝑈𝑦𝑡 (𝑧𝑡 ,𝑑𝑡+1 )= 𝑢𝑡∗ (𝑦𝑡 , 𝑧𝑡 , 𝐸𝑑+1 [𝑈 ∗ (𝑦𝑡,𝑧𝑡 ) ] ) then for p=(𝑦𝑡 ,𝑑𝑡 ) and p’=(𝑦𝑡 ,𝑑′𝑡 ) in 𝑃𝑡 (𝑦𝑡 ) , p≻p’ if and only if ∗ ∗ 𝐸𝑑 [𝑈𝑦𝑡 ] ≽ 𝐸𝑑′ [𝑈𝑦𝑡 ]

4.Temporal Resolution of Uncertainty and Temporal Lotteries Corollary1 : Given a relation ≽ on 𝐷0∗ which satisfies Axioms4.1 4.2 4.3 and functions 𝑈 ∗ and 𝑢𝑡∗ representing ≽ in the sense of Theorem 2 , define 𝑈𝑦𝑇 : 𝑍𝑇 →R by 𝑈𝑦𝑇 (𝑍𝑇 )= 𝑈 ∗ (𝑦𝑇 , 𝑧𝑇 ), and , recursively , define 𝑈𝑦𝑡 : 𝑍𝑡 × 𝑋𝑡+1 →R by (4) 𝑈𝑦𝑡 (𝑧𝑡 ,𝑥𝑡+1 )= max𝑢𝑡∗ (𝑦𝑡 , 𝑧𝑡 , 𝐸𝑑 [𝑈 (𝑦𝑡 ,𝑧𝑡) ])

If binary relations ≽𝑦𝑡 on 𝐷𝑡 are defined by 𝑑𝑡 ≽𝑦𝑡 𝑑𝑡′ if 𝐸𝑑 [𝑈𝑦𝑡 ] ≽ 𝐸𝑑′ [𝑈𝑦𝑡 ] , then the collection {≽𝑦𝑡 }satisfies Axioms2.1 2.2 2.3 3.1.Furthermore, the relations ≽𝑦𝑡 defined by equation(4) are determined by ≽ and do not otherwise depend on the particular functions 𝑈 ∗ and 𝑢𝑡∗ used to represent ≽ . Finally, ≽𝑦0 restricted to 𝐷0∗ coincides with ≽ .

4.Temporal Resolution of Uncertainty and Temporal Lotteries Corollary2 : Given relations ≽𝑦𝑡 on the sets 𝐷𝑡 which satisfy Axiom2.1 2.2 2.3 3.1, if we let ≽ denote the restriction of ≽𝑦0 to 𝐷0∗ , then ≽ satisfies Axioms 4.1 4.2 4.3. Furthermore, if functions 𝑈 ∗ and 𝑢𝑡∗ represent ≽ in the sense of Theorem 2 , and from 𝑈 ∗ and 𝑢𝑡∗ we construct functions 𝑈𝑦𝑡 via equation (4), then the functions 𝑈𝑦𝑡 represents the relations ≽𝑦𝑡 in the sense of Theorem 1.

5.Preferences for earlier or later resolution of Uncertainty Theorem 3 : Suppose the individual’s choice behavior obeys Axioms 2.1 2.2 2.3 3.1 and , as guaranteed by Theorem 1 , his choice behavior is represented by functions U and 𝑢𝑡 . Construct {𝑈𝑦𝑇 }and , for each t , 𝑦𝑡 ,and , 𝑧𝑡 , let Γ(𝑦𝑡 , 𝑧𝑡 )={γ∈R : 𝑈 (𝑦𝑡 ,𝑧𝑡) (𝑥𝑡+1 )for some 𝑥𝑡+1 ∈ 𝑋𝑡+1 } .(the set Γ is the set of values γ which are relevant for 𝑢𝑡 ) Then for fixed t
6.Discussion

The feature : focus on the temporal aspect of uncertainty . Uncertainty is dated . An application of cardinal utility theory to this expanded conception mixture space.